Finding linear functions that maximize AUC scores is important in ranking research. A typical approach to the ranking problem is to reduce it to a binary classification problem over a new instance space, consisting of all pairs of positive and negative instances. Specifically, this approach is formulated as hard or soft margin optimization problems over pn pairs of p positive and n negative instances. Solving the optimization problems directly is impractical since we have to deal with a sample of size pn, which is quadratically larger than the original sample size p + n. In this paper, we reformulate the ranking problem as variants of hard and soft margin optimization problems over p+n instances. The resulting classifiers of our methods are guaranteed to have a certain amount of AUC scores.